We show that there exists a non-archimedean
Fr\'echet-Montel space $W$ with a basis and with a continuous norm
such that any non-archimedean Fr\'echet space of countable type is isomorphic
to a quotient of $W$. We also prove that any non-archimedean nuclear
Fr\'echet space is isomorphic to a quotient of some non-archimedean nuclear
Fr\'echet space with a basis and with a continuous norm.